to become a lieutenant you have to be chosen by a captain. and you normally have to have a certain amount of skill. those who are good in all shinigami arts are preferable, but some captains look for special qualities. for example, the captain of squad 11 will never accept someone skilled in kido, and would readily take one who excels in pure fighting skill, whereas the captain of squad 2 may prefer a person with a certain amount of finesse.
to become a captain there are three ways.
1 pass the captains exam in front of three other captains, including the captain commander. this will involve you showing your bankai, and proving your control over it.
2 receive recommendation from 6 captains, and have three over captains approve it. this does not technically require bankai.
3 defeating a captain one on one, in front of a minimum of 200 other shinigami. yet again, does not need bankai.
numbers 2 and 3 are the hardest to do, because beating a man with bankai is normally difficult, and essentially getting 9 recommendations is also quite an achievement.